Correlation Coefficient Comparisons

A scatter diagram shows paired values of two variables: each point represents one pair. A correlation coefficient summarises an aspect of their relationship using a number between $-1$ and $1$.

Spearman’s rank correlation coefficient and Pearson’s product moment correlation coefficient answer different questions. Spearman’s looks at the order of the values; Pearson’s looks at how closely the points follow a straight line.

Spearman’s: do the orders agree?

A rank is a value’s position when the values are put in order. Suppose five pupils take two tests. If the pupil with the lowest mark in the first test also has the lowest mark in the second, the next-lowest pupil is also next-lowest, and so on, the two rank orders agree perfectly.

Spearman’s coefficient is then $r_s=1$. The gaps between marks do not have to match: it is the pupils’ positions in the two orders that matter.

If the orders are exactly reversed, so that the lowest in one test is the highest in the other and so on, $r_s=-1$. Values nearer either extreme indicate stronger rank-order correlation.

Pearson’s: how closely do the points follow a straight line?

Pearson’s coefficient uses the numerical values rather than just their ranks. It measures the strength and direction of a linear relationship.

For points lying exactly on an upward-sloping straight line, $r=1$. For points lying exactly on a downward-sloping straight line, $r=-1$. Values nearer zero indicate weaker linear correlation.

An increasing pattern need not be straight. If one variable keeps increasing as the other increases, but the increases become progressively larger, the points can follow a curve. The rank orders may still agree perfectly even though the relationship is not perfectly linear.

The same ranks, different shapes

Consider these two constructed sets of paired values. In both, $x$ takes the values 1, 2, 3, 4 and 5. In the first, the corresponding $y$ values are 1, 2, 3, 4 and 5; in the second, they are 1, 2, 4, 8 and 16.

Data for Matching rank orders do not guarantee a straight line

Read each set from left to right. Every successive point is higher than the last, so both sets have perfectly matching rank orders: $r_s=1$.

The first set also lies exactly on a straight line, so $r=1$. The second bends upwards, so its Pearson coefficient is positive but less than 1. Perfect rank-order correlation does not guarantee perfect linear correlation.

Spearman’s does not describe every possible curved relationship. For example, a U-shaped pattern first decreases and then increases, so it does not follow one consistent rank order. Both coefficients can be near zero despite a clear curved relationship. Looking at the scatter diagram helps you understand what the coefficient is summarising.